True and apparent motion of optomechanical resonators, with applications to feedback cooling of gravitational wave detector test masses

Modern optomechanical systems employ increasingly sophisticated quantum-mechanical states of light to probe and manipulate mechanical motion. Squeezed states are now used routinely to enhance the sensitivity of gravitational-wave interferometers to small external forces, and they are also used in feedback-based trapping and damping experiments on the same interferometers to enhance the achievable cooling of fluctuations in the differential test mass mode (arXiv:2102.12665). In this latter context, an accurate accounting of the true test mass motion, incorporating all sources of loss, the effect of feedback control, and the influence of classical force and sensing noises, is paramount. We work within the two-photon formalism to provide such an accounting, which extends a previously described decomposition of the quantum-mechanical noise of the light field (arxiv:2105.12052). This decomposition provides insight, rooted in physically motivated parameters, into the optimal squeezed state and feedback control configuration that should be employed to achieve the lowest fluctuations. We apply this formalism to feedback damping experiments in current and possible future gravitational-wave interferometers – LIGO A+, LIGO Voyager, Cosmic Explorer (CE), and CE Voyager – and discuss how these multi-degree-of-freedom systems might be compared to a single degree-of-freedom oscillator. We find that, for the oscillator definition used most commonly in the literature so far, occupation numbers below 1 are possible in these interferometers over a frequency range comparable to the bandwidth of the trapped and cooled oscillator. We also discuss several technical issues in cooling experiments with gravitational-wave detectors

💡 Research Summary

This paper presents a comprehensive theoretical framework for distinguishing the true mechanical motion of test masses in optomechanical resonators from the apparent motion inferred from optical readout, a distinction that is crucial for evaluating feedback‑based cooling schemes in gravitational‑wave (GW) detectors. Using the two‑photon formalism, the authors model the full set of quantum‑optical fields entering, circulating within, and exiting a complex interferometer, explicitly accounting for optical losses, squeezed‑vacuum injection, filter‑cavity rotation, and measurement‑based feedback control.

The central results are two spectral‑density expressions: Eq. (16) gives the true displacement spectrum Sₓ(Ω) of a test‑mass degree of freedom that is both trapped and cooled by feedback, while Eq. (17) provides the apparent displacement spectrum S_y(Ω) that appears in the homodyne readout. These formulas incorporate all noise sources—external forces, sensing (shot) noise, radiation‑pressure (back‑action) noise, and the correlations that lead to “noise squashing.” By decomposing the quantum noise into physically motivated components (loss‑induced vacuum, fundamental phase noise, and the rotated squeezed state), the authors clarify how each contributes to the true motion.

A key insight is that injecting a squeezed state can indeed reduce the quantum‑noise contribution to the true motion, but the optimal squeezing angle for the true motion differs from that for the readout. Consequently, the frequency‑dependent squeezing produced by a filter cavity—standard in current GW detectors for broadband sensitivity improvement—does not help, and may even be detrimental, for the purpose of minimizing true motion. Moreover, the feedback loop itself generates an effective phase‑noise term that limits the usable amount of squeezing; this noise originates from the lossy effective susceptibility introduced by velocity‑damped feedback, not from optical loss.

The authors define an effective mode occupation number n_eff for the cooled degree of freedom and apply the formalism to several planned detector configurations: LIGO A+, LIGO Voyager, Cosmic Explorer (CE), and CE Voyager. Simulations show that, when operated at design sensitivity, all these interferometers can achieve n_eff < 1 over a frequency band comparable to the trap bandwidth (a few to a few tens of hertz). For example, LIGO A+ can reach sub‑phonon occupation between ~10 Hz and ~30 Hz, while CE can maintain it across a broader 5–50 Hz range.

Finally, the paper discusses practical technical challenges that must be addressed to realize such cooling. These include decoupling the differential arm length degree of freedom from other optical degrees of freedom (power‑recycling, signal‑extraction cavities), mitigating local gravity gradient noise at the test masses, and ensuring low‑noise, low‑latency feedback actuation (electrostatic or magnetic). The authors argue that with appropriate modifications to the sensing and control architecture, feedback cooling below the quantum ground state is feasible in current and next‑generation GW detectors, opening the door to quantum‑state preparation of macroscopic test masses and enabling novel fundamental‑physics experiments such as probing gravitationally‑mediated entanglement.